Problem 3 - Algebra and Trigonometry

Question

The average rate of change of the function \(f(x)=x^{2}\) between \(x=1\) and \(x=5\) is average rate of change \(=\frac{\square}{\square}=\)_____.

Step-by-Step Solution

Answer

The average rate of change is 6.

Step 1: Understand the Formula

The average rate of change of a function between two points \(x = a\) and \(x = b\) is calculated using the formula \(\frac{f(b) - f(a)}{b - a}\). Here, \(f(x) = x^2\), \(a = 1\), and \(b = 5\).

Step 2: Calculate \(f(1)\)

Substitute \(x = 1\) into the function: \(f(1) = 1^2 = 1\).

Step 3: Calculate \(f(5)\)

Substitute \(x = 5\) into the function: \(f(5) = 5^2 = 25\).

Step 4: Apply the Average Rate of Change Formula

Substitute \(f(1)\), \(f(5)\), \(a = 1\), and \(b = 5\) into the average rate of change formula: \(\frac{f(5) - f(1)}{5 - 1}\). This results in \(\frac{25 - 1}{4}\).

Step 5: Simplify the Expression

Calculate \(25 - 1 = 24\), then divide by \(4\): \(\frac{24}{4} = 6\). Thus, the average rate of change is 6.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental part of algebra and represent parabolic graphs. These functions are typically in the form of \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a eq 0\). A parabolic graph opens upwards if \(a > 0\) and downwards if \(a < 0\).
The quadratic function in our exercise is \(f(x) = x^2\). This is a simple case where \(a = 1\), \(b = 0\), and \(c = 0\). Its graph is a classic "U" shape, symmetric about the y-axis. This symmetry occurs because of the absence of the linear term \(b\) and term \(c\), keeping the parabola centered at the origin.
In calculations involving quadratic functions, especially when determining rates of change, we are interested in how the value of \(f(x)\) changes as \(x\) changes over an interval. In this exercise, the average rate of change indicates how steep or gentle the graph's slope is between two points on this parabola.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations, forming the basis for algebraic equations and functions. In understanding quadratic functions like \(f(x) = x^2\), recognizing these expressions helps make calculations easier.
When calculating the average rate of change, an algebraic expression functions as a tool to simplify problems. For example, \(f(x)=x^2\) results in specific expressions when values for \(x\) are plugged in. For \(x = 1\), \(f(1) = 1^2 = 1\); for \(x = 5\), \(f(5) = 25\). These expressions are calculated step by step to find particular values that help analyze the function's behavior between given points.
Breaking complex expressions down into smaller parts or calculations can reveal insights into what the function does between specific values. Here, understanding and simplification are key aspects that underline the theoretical and practical use of algebra in problem-solving.

Function Analysis

Function analysis involves a comprehensive examination of a function's behavior over a particular interval or domain. It's crucial when determining something like the average rate of change.
The average rate of change gives an overall picture of a function's growth or decline over a specific segment. It's essentially a slope of the line connecting two points on a function graph. In our scenario, between \(x = 1\) and \(x = 5\), we calculated this with the formula \(\frac{f(b) - f(a)}{b - a}\). Substituting values \(\frac{25 - 1}{5 - 1}\) we found the result was 6. This rate tells us, on average, how steep the curve is between these points.
Function analysis using this approach offers insights into how the function behaves between specific values compared to just observing singular points. This method can be applied to many polynomial functions, not just quadratic ones, facilitating a deeper understanding of their behavior over chosen intervals.